PHYSICAL REVIEW X 5, 021017 (2015)
Magnetic End States in a Strongly Interacting One-Dimensional
Topological Kondo Insulator
Alejandro M. Lobos,1,2,* Ariel O. Dobry,1 and Victor Galitski2,3
1
Facultad de Ciencias Exactas Ingeniería y Agrimensura, Universidad Nacional de Rosario and Instituto
de Física Rosario, Boulevard 27 de Febrero 210 bis, 2000 Rosario, Argentina
2
Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
3
School of Physics, Monash University, Melbourne, Victoria 3800, Australia
(Received 4 December 2014; revised manuscript received 10 April 2015; published 22 May 2015)
Topological Kondo insulators are strongly correlated materials where itinerant electrons hybridize with
localized spins, giving rise to a topologically nontrivial band structure. Here, we use nonperturbative
bosonization and renormalization-group techniques to study theoretically a one-dimensional topological
Kondo insulator, described as a Kondo-Heisenberg model, where the Heisenberg spin-1=2 chain is coupled to
a Hubbard chain through a Kondo exchange interaction in the p-wave channel (i.e., a strongly correlated
version of the prototypical Tamm-Schockley model). We derive and solve renormalization-group equations at
two-loop order in the Kondo parameter, and find that, at half filling, the charge degrees of freedom in the
Hubbard chain acquire a Mott gap, even in the case of a noninteracting conduction band (Hubbard parameter
U ¼ 0). Furthermore, at low enough temperatures, the system maps onto a spin-1=2 ladder with local
ferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-1 chain
with frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypical
interacting topological spin model, and features two magnetic spin-1=2 end states for chains with open
boundary conditions. Our analysis allows us to derive an insightful connection between topological Kondo
insulators in one spatial dimension and the well-known physics of the Haldane chain, showing that the ground
state of the former is qualitatively different from the predictions of the naive mean-field theory.
DOI: 10.1103/PhysRevX.5.021017
Subject Areas: Condensed Matter Physics,
Strongly Correlated Materials,
Topological Insulators
I. INTRODUCTION
Starting with the pioneering works of Kane and Mele
[1,2] and others [3–5], there has been a surge of interest
in topological characterization of insulating states [6–8].
It is now understood that there exist distinct symmetryprotected classes of noninteracting insulators, such that
two representatives from different classes cannot be
adiabatically transformed into one another (without closing
the insulating gap and breaking the underlying symmetry
along the way). A complete topological classification
of such band insulators has been developed in the form
of a “periodic table of topological insulators” [9,10].
Furthermore, it was realized that the nontrivial (topological)
insulators from this table possess, as their hallmark features,
gapless boundary modes. The latter have been spectacularly
*
lobos@ifir‑conicet.gov.ar
Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
2160-3308=15=5(2)=021017(13)
observed in a variety of experiments in both three- [11,12]
and two-dimensional systems [13–16].
The aforementioned classification, however, is limited to
noninteracting systems, and as such, it represents a classification of single-particle band structures. Adding interactions to the theory leads to significant complications.
Understanding and classifying strongly interacting topological insulator phases in many-particle systems is a
fundamental open problem in condensed matter.
A class of material that combines strong interactions and
nontrivial topology of emergent bands is topological Kondo
insulators (TKI) [17]. A basic model of these heavy
fermion systems involves even-parity conduction electrons
hybridizing with strongly correlated f electrons. At low
temperatures, a hybridization gap opens up and an insulating state can be formed. Its simplified mean-field description makes it amenable to a topological classification
according to the noninteracting theory, and a topologically
nontrivial state appears due to the opposite parities of the
states being hybridized. Although the mean-field description (formally well controlled in the large-N approximation
[18–20]) does appear to correctly describe the nature of the
topological Kondo insulating states observed in bulk
021017-1
Published by the American Physical Society
LOBOS, DOBRY, AND GALITSKI
PHYS. REV. X 5, 021017 (2015)
materials thus far [21–25], it is interesting to see if
nonperturbative effects beyond mean field can qualitatively
change the mean-field picture.
In contrast to higher dimensions, where reliable theoretical techniques to treat strong interactions are scarce,
there exists a rich arsenal of such nonperturbative methods
for one-dimensional systems, where strongly correlated
“non-mean-field” ground states abound. Since the Kondo
insulating Hamiltonian and its mean-field treatment are
largely dimension independent, it is interesting to consider
the one-dimensional model as a natural playground to study
the interplay between strong interactions and nontrivial
topology.
With this motivation in mind, we study here a strongly
interacting model of a one-dimensional topological Kondo
insulator, i.e., a “p-wave” Kondo-Heisenberg model, introduced earlier by Alexandrov and Coleman [26], who
treated the problem in the mean-field approximation.
Here, we go beyond the mean-field level and consider
quantum fluctuations nonpeturbatively, using the Abelian
bosonization technique. It is shown that a “topological
coupling” between the electrons in the Hubbard chain and
spins in the Heisenberg chain gives rise to a charge gap at
half filling in the former. The relevant interaction between
the remaining spin-1=2 degrees of freedom in the chains is
effectively ferromagnetic, which locks them into a state
qualitatively similar to the Haldane spin-1 chain. The
ground state, therefore, is a strongly correlated topological
insulator, which exhibits neutral spin-1=2 end modes.
While our main motivation is essentially theoretical
(i.e., to allow a deeper understanding of strongly interacting
topological matter), we believe our results might have
direct application in ultracold atom experiments, where
double-well optical superlattices loaded with atoms in s and
p orbitals have been realized [27,28]. In addition, our work
might have some relevance in recent experimental results
[29–31], which suggest the existence of a ferromagnetic
phase transition and/or suppressed surface charge transport
in samples of samarium hexaboride (SmB6 —a threedimensional topological Kondo insulator).
This article is organized as follows. In Sec. II, we specify
the model for a 1D TKI and introduce the Abelian
bosonization description. In Sec. III, we present the
renormalization-group (RG) analysis and discuss the quantum phase diagram of the system. In Sec. IV, we analyze the
topological aspects of the problem and explain the emergence of topologically protected magnetic edge states, and
in Sec. V, we present a summary and discussion of the
results. Finally, in the Appendix, we present the technical
derivation of the renormalization-group equations.
II. MODEL
We start our theoretical description by considering
the Hamiltonian of the system depicted in Fig. 1,
H ¼ H 1 þ H2 þ H K , where
FIG. 1. Schematic representation of the 1D “p-wave” KondoHeisenberg model. The top chain corresponds to the Hubbard
model and the bottom chain is the S ¼ 1=2 antiferromagnetic
Heisenberg model. The Kondo exchange (depicted pictorially as
slanted bonds with odd inversion symmetry) is a nonlocal
nearest-neighbor antiferromagnetic interaction J K > 0 that
couples a spin Sj in the Heisenberg chain with the p-wave spin
density πj in the Hubbard chain [see Eq. (4)].
H1 ¼ −t
N
s −1
X
Ns
X
ðc†j;σ cjþ1;σ þ H:c:Þ − μ
j¼1;σ
nj;σ
j¼1;σ
Ns X
1
1
nj↑ −
nj↓ −
þU
2
2
j¼1
ð1Þ
is a fermionic 1D Hubbard chain with N s sites, where
nj;σ ¼ c†j;σ cj;σ is the density of spin-σ electrons at site j, μ is
the chemical potential, and U is the Hubbard interaction
parameter. In this work, we focus on only the half filled
case μ ¼ 0, where there is one electron per site. However,
we expect our results to also remain valid for small
deviations of half filling. The spin chain is described by
the spin-1=2 Heisenberg model,
H2 ¼ J
N
s −1
X
Sj · Sjþ1 ;
ð2Þ
j¼1
with J > 0. Here, we assume the same lattice parameter a
for both chains H1 and H2 . Finally, motivated by the work
by Alexandrov and Coleman [26], we assume the following
exchange coupling between the two chains,
HK ¼ JK
Ns
X
Sj · π j ;
ð3Þ
j¼1
where JK > 0 is the Kondo interaction between the jth
spin (Sj ) in the Heisenberg chain and the p-wave spin
density in the fermionic chain at site j, defined as
πj ≡
p†j;α
σαβ
pj;β ;
2
ð4Þ
pffiffiffi
where pj;α ≡ ðcjþ1;α − cj−1;α Þ= 2 is a linear combination
of orbitals with p-wave symmetry and σαβ is the vector of
Pauli matrices. This model can be regarded as a strongly
interacting version of the Tamm-Shockley model [32–34].
While for our present purposes this is an interesting
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MAGNETIC END-STATES IN A STRONGLY-INTERACTING …
“toy model” Hamiltonian that allows us to extract a useful
insight into strongly interacting topological phases, it
could, in principle, be realized in ultracold-atom experiments (see Sec. V for details). In the absence of interactions
in the fermionic chain (i.e., U ¼ 0) and in the large-N
mean-field approximation, Alexandrov and Coleman have
shown the emergence of topologically protected edge states
arising from the nontrivial form of the Kondo term [Eq. (3)]
[26]. In their mean-field approach, the effective description
of the system corresponds to noninteracting quasiparticles
filling a strongly renormalized valence band with a nontrivial topology, stemming from the charge conjugation,
time reversal, and charge U(1) symmetry of the effectively
noninteracting Hamiltonian (see also Ref. [35] for a
discussion of a closely related system).
In this paper, our goal is to understand the emergence of
topologically protected edge states without introducing any
decoupling of the Kondo interaction, including the interacting case, U ≠ 0. We consider the case of small U and
JK . This is formally represented by linearizing the noninteracting spectrum ϵk ¼ −2t cos ka in the fermionic
chain H 1 around the Fermi energy μ ¼ 0 and taking the
continuum limit where the lattice constant a → 0. Then, the
fermionic operators admit the low-energy representation
[36,37]
cj;σ
pffiffiffi ∼ eikF xj R1;σ ðxj Þ þ e−ikF xj L1;σ ðxj Þ;
a
ð5Þ
where R1;σ ðxÞ and L1;σ ðxÞ are right- and left-moving
fermionic field operators, which vary slowly on the scale
of a. As we are interested in the edge-state physics, we
consider open boundary conditions c0;σ ¼ cN s þ1;σ ¼ 0,
leading to the following constraints:
L1;σ ð0Þ ¼ −R1;σ ð0Þ;
ð6Þ
L1;σ ðLc Þ ¼ −ei2kF Lc R1;σ ðLc Þ;
ð7Þ
where Lc ¼ N s a is the length of the chain. We next
introduce the Abelian bosonization formalism [36,37],
F1;σ −iϕ1;R;σ ðxÞ
R1;σ ðxÞ ¼ pffiffiffiffiffiffiffiffi
;
e
2πα
F1;σ iϕ1;L;σ ðxÞ
L1;σ ðxÞ ¼ pffiffiffiffiffiffiffiffi
;
e
2πα
ð8Þ
PHYS. REV. X 5, 021017 (2015)
constraints (6) and (7) introduced by the open boundary
conditions, the right and left movers are not independent,
and obey the constraints
ϕ1;L;σ ð0Þ ¼ −ϕ1;R;σ ð0Þ þ π;
ϕ1;L;σ ðLc Þ ¼ −ϕ1;R;σ ðLc Þ þ 2kF Lc − π þ 2qσ π:
ð9Þ
ð10Þ
Here, qσ is an integer representing the occupation of the
“zero-mode” excitations, i.e., particle-hole excitations with
momentum k ¼ 0 and total spin σ. Its presence in Eq. (10)
can be understood recalling that the expression of the
nonchiral bosonic field ϕ1;σ ¼ ðϕ1;R;σ þ ϕ1;L;σ Þ=2 is [38]
π
x
þ ðkF Lc − π þ πqσ Þ
2
Lc
∞
X
sin ðkn xÞ
pffiffiffi ðαn;σ þ α†n;σ Þ;
þ
n
n¼1
ϕ1;σ ðxÞ ¼
where kn ≡ πn=Lc , with integer n > 0, and α†n;σ are
bosonic operators obeying the commutation relation
½αn;σ ; α†m;σ 0 ¼ δn;m δσ;σ0 (see Refs. [38,39] for details).
From here, we obtain the additional commutation
relations [38]
8
>
< −iπδσ;σ 0
½ϕ1;Rσ ðxÞ; ϕ1;Lσ 0 ðyÞ ¼ 0
>
:
−2πiδσ;σ0
for 0 < x; y < Lc
for x ¼ y ¼ 0
for x ¼ y ¼ Lc :
ð11Þ
The bosonization procedure applied to the 1D Hubbard
model is standard, and we refer the reader to textbooks for
details [36,37].
pffiffiffi Introducing charge and spin bosonic fields
ϕ1;λσ ¼ 1= 2½ϕ1c − λθ1c þ σðϕ1s − λθ1s Þ (where the convention of signs λ ¼ fR; Lg ¼ fþ; −g and σ ¼ f↑; ↓g ¼
fþ; −g is implied), the 1D Hubbard model at half filling
(i.e., kF ¼ π=2a) becomes [36]
X Z Lc v1ν
v1ν K 1ν
2
2
H1 ¼
dx
ð∂ ϕ Þ þ
ð∂ x θ1ν Þ
2πK 1ν x 1ν
2π
ν¼c;s 0
Z L
pffiffiffi
pffiffiffi
c
U
dx½cosð 8ϕ1c Þ − cosð 8ϕ1s Þ; ð12Þ
−
2
2ðπaÞ 0
where the new fields obey the boundary conditions
where α is a short distance cutoff in the bosonization
procedure (we take α ¼ a hereafter). In Eq. (8), ϕ1;λσ ðxÞ
(with λ ¼ fR; Lg) are bosonic fields obeying the commutation relations ½ϕ1;Rσ ðxÞ; ϕ1;Rσ 0 ðyÞ ¼ iπsgnðx − yÞδσ;σ0 ,
½ϕ1;Lσ ðxÞ; ϕ1;Lσ0 ðyÞ ¼ −iπsgnðx − yÞδσ;σ 0 , and F1;σ are
Klein operators, which obey anticommutation relations
fF1;σ ; F1;σ 0 g ¼ δσ;σ 0 , and, therefore, ensure the correct
anticommutation relations for fermions. Because of the
021017-3
ϕ1s ð0Þ ¼ 0;
π
ϕ1s ðLc Þ ¼ pffiffiffi ðq↑ − q↓ Þ;
2
ð13Þ
π
ϕ1c ð0Þ ¼ pffiffiffi ;
2
pffiffiffi
π
π
ϕ1c ðLc Þ ¼ 2 kF Lc −
þ pffiffiffi ðq↑ þ q↓ Þ; ð14Þ
2
2
LOBOS, DOBRY, AND GALITSKI
PHYS. REV. X 5, 021017 (2015)
and the commutation relation ½ϕ1;ν ðxÞ; θ1;ν0 ðyÞ ¼
−iðπ=2Þδν;ν0 sgnðx − yÞ. From here, we conclude that the
field ð1=πÞ∂ x θ1;ν ðxÞ is the momentum canonically conjugated to ϕ1;ν ðxÞ.
As is well known, in 1D charge and spin excitations
generally decouple and the above Hamiltonian can be split
as H 1 ¼ H 1c þ H 1s , with the first line describing independent Luttinger liquids for the charge and spin sectors,
which are characterized p
byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
charge (spin)ffi acoustic modes
with velocities v1c ¼ vF 1 þ Ua=ðπvF Þ ðv1s ¼ vF Þ, and
Luttinger parameterp
controlling
the decay
of the correlation
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
functions K 1c ¼ 1= 1 þ Ua=ðπvF Þ (K 1s ¼ 1). The presence of the cosine terms in the second line of Eq. (12)
changes the physics qualitatively. In the present work, we
restrictpour
ffiffiffi focus to the case U ≥ 0, where the term
∼ cosð 8ϕ1c Þ is marginally relevant in the renormalization-group sense and opens a Mott gap
pffiffiin
ffi the charge sector.
At the same time, the term ∼ cosð 8ϕ1s Þ is marginally
irrelevant at the SU(2) symmetric point, and the spin sector
remains gapless [36,37].
The bosonization of the Heisenberg chain H2 is also
quite standard, and we refer the reader to the abovementioned textbooks [36,37]. A usual trick consists of
representing the spin operators Sj by auxiliary fermionic
operators in a half filled Hubbard model with interaction
parameter U 0 ≫ U. Therefore, while technically the procedure is identical to Eq. (12), the charge degrees of
freedom in the bosonized Hamiltonian, H 2 can be assumed
to be absent at the relevant energy scales of the problem due
to the Mott gap ∼U 0 . Then, ignoring the charge degrees of
freedom and irrelevant operators in Eq. (12), and replacing
the chain label 1 → 2, we obtain
Z L
c
v
H2 ¼ 2s
dx½ð∂ x ϕ2s Þ2 þ ð∂ x θ2s Þ2 :
ð15Þ
2π 0
Finally, we bosonize the Kondo Hamiltonian. The
p-wave spin density in the fermionic chain and the spin
density in the Heisenberg chain are, respectively,
πj
∼ 2½J1R ðxj Þ þ J1L ðxj Þ − ð−1Þj N1 ðxj Þ;
a
Sj
∼ J2R ðxj Þ þ J2L ðxj Þ þ ð−1Þj N2 ðxj Þ;
a
pffiffiffi
1
cosf 2½λϕa;s ðxÞ − θa;s ðxÞg;
2πa
pffiffiffi
1
sinf 2½λϕa;s ðxÞ − θa;s ðxÞg;
Jyaλ ðxÞ ¼
2πa
Jxaλ ðxÞ ¼
1
Jzaλ ðxÞ ¼ − pffiffiffi ½∂ x ϕa;s ðxÞ − λ∂ x θa;s ðxÞ;
8π
ð18Þ
ð19Þ
ð20Þ
where λ ¼ RðLÞ corresponds to the plus (minus) sign, and
where Na ðxÞ ¼ R†a;α ðxÞðσα;β =2ÞLa;β ðxÞ þ H:c: are the staggered components
pffiffiffi
pffiffiffi
pffiffiffi
m2
fcos½ 2θ2s ðxÞ; − sin½ 2θ2s ðxÞ; − cos½ 2ϕ2s ðxÞg;
πa
pffiffiffi
with m2 ¼ hcos ½ 2ϕ2c ðxÞi resulting from the integration
of the gapped charge degrees of freedom in the Heisenberg
chain. Therefore, although the spin densities (16) and (17)
look similar in the bosonized language, they actually differ
in two crucial aspects. (1) While in Eq. (22) the charge
degrees of freedom are absent in the expression of the
staggered magnetization, they are still present in Eq. (21) in
pffiffiffi
the term cos½ 2ϕ1c ðxÞ and we need to consider them.
(2) Comparing Eqs. (16) and (17), we note a sign difference
in the staggered components. This sign is related to the
p-wave nature of the operators pj;σ , and is, therefore,
intimately connected to the topology of the Kondo interaction. The role of this sign turns out to be crucial in the rest
of the paper.
ð17Þ
where JaR ðxÞ ¼ R†a;α ðxÞðσα;β =2ÞRa;β ðxÞ and JaL ðxÞ ¼
L†a;α ðxÞðσα;β =2ÞLa;β ðxÞ
(with
a ¼ 1; 2)
are
the
smooth components of the spin density, with bosonic
representation
pffiffiffi
pffiffiffi
pffiffiffi
pffiffiffi
cos ½ 2ϕ1c ðxÞ
N1 ðxÞ ¼
fcos½ 2θ1s ðxÞ; − sin½ 2θ1s ðxÞ; − cos½ 2ϕ1s ðxÞg;
πa
N2 ðxÞ ¼
ð16Þ
ð21Þ
ð22Þ
Replacing the above results into Eq. (3), and taking the
continuum limit, the Kondo interaction becomes, in the
bosonic language,
Z
H K ∼ 2JK a
0
Lc
dx
X
λ;λ0 ¼L;R
− ∶N1 ðxÞ · N2 ðxÞ∶ ;
∶J1λ ðxÞ · J2λ0 ðxÞ∶
ð23Þ
where the sign in the second line is a consequence of the
above-mentioned sign in the staggered part of πðxÞ.
Note that this model is reminiscent of the (nontopological) 1D Kondo-Heisenberg model, which has recently
021017-4
MAGNETIC END-STATES IN A STRONGLY-INTERACTING …
received much attention in the context of pair-density
wave-ordered phases in high-T c cuprate physics [40–46],
and of the Hamiltonian of a spin-1=2 ladder [38,47–49].
However, a crucial difference from those works is the
nontrivial structure of the Kondo interaction, which
differs from the usual coupling ∼JK Sj · sj , where sj ≡
c†j;α ðσαβ =2Þcj;β is the standard (i.e., s wave in this context)
spin density in the fermionic chain. The first line in Eq. (23)
is, in fact, closely related to the model considered in
Refs. [41–45]. In the half filling situation we are analyzing
here, however, the most relevant part of HK (in the RG
sense) is given by the product N1 ðxÞ · N2 ðxÞ, which
dominates the physics at low energies [38,47–49]. The
term N1 ðxÞ · N2 ðxÞ survives when the system is at (or close
enough to) half filling, and when both chains have the same
lattice parameter (in other situations, the oscillatory factors
ei2kF x suppress this term, and the situation corresponds to
the case analyzed in Refs. [41–45]). Therefore, for our
present purposes, we can neglect the first term in Eq. (23)
and focus on the second term:
Z
HK ≈ −2JK a
Lc
0
Z
J m
HK ¼ − K2 2
π a
Z
Lc
0
pffiffiffi
dx cosð 2ϕ1c Þ
× ½− cos ð2ϕþ Þ þ cos ð2ϕ− Þ þ 2 cos ð2θ− Þ:
ð25Þ
In what follows, we assume identical spinon dispersion
v1s ¼ v2s ¼ vs . Although this assumption is certainly an
idealization, one can show that the asymmetry δv ≡
v1s − v2s is a marginal perturbation in the renormalization-group sense, and therefore we do not expect that small
asymmetries will have a qualitative effect on our results.
We now write the Euclidean action of the system using
complex space-time coordinates z ¼ vF τ þ ix and z̄ ¼
vF τ − ix, with τ ¼ it the imaginary time, and the left
and right fields ϕν ¼ ðϕνL þ ϕνR Þ=2, where fν ¼ þ; −; 1cg.
The Euclidean action becomes
S ¼ S0 þ SU þ SK ;
ð26Þ
Z
1
2
d r ð∂ z ϕcL Þ2 þ ð∂ z̄ ϕcR Þ2
S0 ¼ −
4π
X
2
2
þ
½ð∂ z ϕνR Þ þ ð∂ z̄ ϕνR Þ ;
ð27Þ
ν¼
dxN1 ðxÞ · N2 ðxÞ;
pffiffiffi
Lc
2J m
¼ − K2 2
dx cosð 2ϕ1c Þ
π a 0
pffiffiffi
pffiffiffi
pffiffiffi
× ½cosð 2θ1s Þ cosð 2θ2s Þ þ sinð 2θ1s Þ
pffiffiffi
pffiffiffi
pffiffiffi
× sinð 2θ2s Þ þ sinð 2ϕ1s Þ sinð 2ϕ2s Þ:
PHYS. REV. X 5, 021017 (2015)
Z
SU ¼ G2c
d2 rO2c ðrÞ þ G3
Z
SK ¼ GK
ð24Þ
At this point, we note that the problem is reminiscent of the
well-known case of S ¼ 1=2 ladders with open boundary
conditions [38,48], with the important
difference that here
pffiffiffi
there is an extra factor ∼ cosð 2ϕ1c Þ.
The physics of the spin sector [i.e., the term in square
brackets in Eq. (24)] is quite nontrivial due to the presence
of both the canonically conjugate fields ϕa;s ðxÞ and θa;s ðxÞ,
which cannot be simultaneously stabilized. However,
the analysis of the charge sector is simpler, as only the
field ϕ1c ðxÞ appears in the expression. This means that
in the limit JK → ∞, the system can gain energy by
“freezing
pffiffiffi out” the charge degrees of freedom, i.e., m1 ¼
hcos½ 2ϕ1c ðxÞi, as there is no other competing mechanism. In the next section, we substantiate these ideas by
providing a rigorous analysis.
III. RENORMALIZATION-GROUP ANALYSIS
Based on the similarity with the physics of spin ladders,
we p
introduce
symmetric and antisymmetric
fields ϕ ¼
ffiffiffi
pffiffiffi
ð1= 2Þðϕ1s ϕ2s Þ and θ ¼ ð1= 2Þðθ1s θ2s Þ, in terms
of which the Hamiltonian becomes
Z
d2 rO3 ðrÞ;
d2 r
pffiffiffi OK ðrÞ;
a
ð28Þ
ð29Þ
with d2 r ¼ vF dxdτ, and where we have defined the
dimensionless couplings,
G02c ¼ G03 ¼
Ua
;
πvF
G0K ¼
JK m2 a
;
πvF
ð30Þ
and the scaling operators,
1
∂ ϕ ∂ ϕ ;
4π z 1cL z̄ 1cR
ð31Þ
pffiffiffi
2π
∶
cosð2
2ϕ1c Þ∶;
L2
ð32Þ
O2c ¼
O3 ¼ −
pffiffiffiffiffiffi
pffiffiffi
2π
OK ¼ − 3=2 ∶ cosð 2ϕ1c Þ½− cos ð2ϕþ Þ
L
þ cos ð2ϕ− Þ þ 2 cos ð2θ− Þ∶;
ð33Þ
where we have explicitly normal ordered the operators. S0
corresponds to a free fixed-point action, and SU and SK are
perturbations arising from the Hubbard repulsion in chain 1
and the Kondo interchain interaction, respectively. We
neglect all the perturbations in the spin sector generated
by U, as they renormalize to zero along the SU(2)-invariant
line in the parameter space. We also neglect the less
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LOBOS, DOBRY, AND GALITSKI
PHYS. REV. X 5, 021017 (2015)
relevant terms coming from the product of the smooth part
of the currents in Eqs. (16) and (17).
Expanding
the
generating
functional
Z¼
RQ
−S½ϕ
;θ
i
i
perturbatively at second order
i¼f1c;g D½ϕi ; θ i e
in Gi , we obtain the product of the different operators [i.e.,
the operator product expansion (OPE)] of Eqs. (31)–(33).
Importantly, the OPE of the Kondo interaction
OK ðz0 ; z̄0 ÞOK ðz; z̄Þ gives rise to operators O3 and O2c ,
which are already present in the charge sector of chain 1.
This corresponds to an effective dynamically generated
Hubbard repulsion originated by the interchain Kondo
coupling (see the Appendix). Therefore, even for an
initially noninteracting chain (i.e., U ¼ 0), this emergent
repulsive interaction induces the opening of a Mott insulating gap in the charge sector of the half filled conduction
band. At energies below this gap, the p
field
ffiffiffi ϕ1c becomes
pinned to the degenerate values 0 or π= 2. Note that only
the latter is consistent with the boundary conditions given
in Eq. (14). Therefore, this analysis suggests that the energy
is minimized by a uniform configurationpof
ffiffiffi the field ϕ1c ,
which “freezes” at the bulk value π= 2. While other
configurations with kink excitations connecting the different minima are certainly possible, these configurations
are more costly energetically speaking and do not belong to
the ground state. Therefore, energy minimization prevents
the ground state from developing “kink” excitations in the
charge sector and, consequently, we can exclude the
presence of localized charge edge states. Then, at low
enough energies, the system becomes a Mott insulator in
the bulk due to the electronic correlations, and no topological effects arise in the charge sector.
A more quantitative study can be done by analyzing the
two-loop RG-flow equations (see the Appendix for details):
cos ð2ϕ Þ ¼ −iπaðη†R ηL − η†L ηR Þ;
ð37Þ
cos ð2θ Þ ¼ iπaðη†R η†L − ηL ηR Þ:
ð38Þ
For later purposes, it is more convenient to introduce a
Majorana-fermion representation of the fields ηþ;λ ¼
pffiffiffi
pffiffiffi
1= 2ðξ2λ þ iξ1λ Þ, η−;λ ¼ 1= 2ðξ3λ þ iξ0λ Þ; the Hamiltonian
can be compactly written as H ¼ H0 þ HK , where
pffiffiffi Z
v1c Lc
ð∂ x ϕ1c Þ2
U cos 8ϕ1c
2
dx
þ K 1c ð∂ x θ1c Þ −
H0 ¼
2π 0
K 1c
v1c πa2
Z
3
Lc
v X
dx½ξaR ∂ x ξaR − ξaL ∂ x ξaL ;
ð39Þ
−i s
2 a¼0 0
J m
HK ¼ i K 2
2π
Z
0
Lc
3
X
pffiffiffi
0 0
a a
dx cosð 2ϕ1c Þ 3ξR ξL −
ξR ξL ;
a¼1
ð40Þ
dG2c
3
¼ G23 þ G2K ;
4
dl
ð34Þ
dG3
3
¼ G2c G3 þ G2K ;
4
dl
ð35Þ
dGK GK 1
¼
þ G2c GK þ G3 GK ;
4
dl
2
G3 ðlÞ ¼ 34 ðG0K Þ2 ðel − 1Þ, from its initial value G3 ð0Þ ¼ 0.
This produces the anticipated Mott insulating gap Δc in the
charge sector. Its dependence with the parameters can be
obtained by the procedure described in Ref. [36] (p. 65).
We obtain Δc ∼ ðG0K Þ2 for small enough JK . We can
envisage that if U and JK would be of the same order,
the dominant contribution to Δc would come from the
interchain Kondo coupling.
Following previous references [38,47–49], we can refermionize Eq. (25), noting that the scaling dimension of the
cosines in the square bracket exactly corresponds to the
free-fermion point and therefore they can be written in
terms of right- and left-moving free Dirac fermion fields
η;R ðxÞ and η;L ðxÞ as
where the Majorana fields obey the boundary conditions
ð36Þ
where l ¼ ln ða=a0 Þ is the logarithmic RG scale. When
GK ¼ 0, these equations reduce to the Kosterlitz-Thouless
RG equations corresponding to the charge sector of the
Hubbard model. They predict a charge gap that is exponentially small in U [36].
Let us now analyze the case of an initially U ¼ 0. In this
situation, only the linear term survives in the right-hand
side of Eq. (36), which expresses the relevance of the
operator OK and gives rise to an exponential increase of
GK ðlÞ with the RG scale l as GK ðlÞ ¼ G0K el=2 . As a
consequence, the coupling G3 ðlÞ in Eq. (35), representing
the Hubbard repulsion, also increases exponentially
ξaR ð0Þ ¼ ξaL ð0Þ;
ð41Þ
ξaR ðLc Þ ¼ ξaL ðLc Þ:
ð42Þ
The uniform symmetric and antisymmetric spin densities in
the ladder become [47]
i
Maλ ¼ J a1;λ þ Ja2;λ ¼ − ϵabc ξbλ ξcλ ;
2
ð43Þ
i
K aλ ¼ Ja1;λ − J a2;λ ¼ − ξ0λ ξaλ ;
2
ð44Þ
with a ¼ f1; 2; 3g and λ ¼ fR; Lg. This is a well-known
representation of two independent SUð2Þ1 Kac-Moody
currents J1;λ and J2;λ in terms of four Majorana fields
[47]. In our case, these four degrees of freedom, resulting
from the combination of the two SU(2) spin-density fields
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MAGNETIC END-STATES IN A STRONGLY-INTERACTING …
PHYS. REV. X 5, 021017 (2015)
in the two chains, are expressed in terms of a singlet ξ0λ and
triplet ξaλ Majorana fields.
From the previous analysis, we conclude that at temperatures T ≪ Δc , the charge and spin degrees of freedom
become effectively decoupled, and the low-energy
~ ¼
Hamiltonian of the system can be written as H → H
~
~
Hc þ Hs , with
ð∂ x ϕ1c Þ2
þ K 1c ð∂ x θ1c Þ2
K
0
1c
U eff m21
þ
ðϕ1c Þ2 ;
v1c πa2
~ c ¼ v1c
H
2π
Z
Lc
dx
ð45Þ
where, based on the discussion above Eq. (34), we
pffiffiffi have
expanded the charge field near the value ϕ1c ¼ π= 2, and
3 Z L
X
c
~ s ¼ − i vs
dx½ξaR ∂ x ξaR − ξaL ∂ x ξaL H
2 a¼0 0
Z
3
X
JK m2 m1 Lc
0 0
a a
þi
dx 3ξR ξL −
ξR ξL ;
2π
0
a¼1
ð46Þ
where U eff ≡ UðJK Þ is the renormalized Coulomb repulsion parameter and where the charge degrees of freedom
in thepffiffifermionic
chain develop a gap m1 ≡ m1 ðJK Þ ¼
ffi
hcosð 2ϕ1c Þi. Note that, in this form, the spin
~ s is similar to that of a spin-1=2 ladder
Hamiltonian H
[38,47,48]. The quantitative determination of the parameter
m1 requires a self-consistent calculation, which is beyond
the scope of the present work. Nevertheless, the previous
analysis allows us to understand two important aspects of
the topological Kondo-Heisenberg chain: (a) the generation
of an insulating state in the bulk (necessary to reproduce the
bulk insulating state of a TKI) and (b) the emergence of
magnetic edge states, which is the subject of the next
section.
IV. S ¼ 1=2 MAGNETIC EDGE MODES
AND TOPOLOGICAL INVARIANT
The previous analysis shows that, at low temperatures
T ≪ Δc , the 1D p-wave Kondo-Heisenberg chain at half
filling can be effectively mapped onto a spin ladder
problem, which is dominated by the staggered components
of the spin densities. Interestingly, for an antiferromagnetic
Kondo coupling JK > 0, the negative sign emerging from
the structure of the nonlocal Kondo interaction (24)
effectively induces a local ferromagnetic interaction [see
vertical dashed lines in Fig. 2(a)]. It is well known that the
spin ladder with ferromagnetic exchange coupling J⊥ < 0
along the rungs has a low-energy triplet sector [38,47,48],
which maps onto the Haldane spin-1 chain [50,51].
Therefore, at temperatures T ≪ J⊥ , our model describes
the physics of the Haldane spin-1 chain, which is known to
FIG. 2. (a) At temperatures T ≪ Δc ∼ J 2K , the dynamically
induced Mott gap, the charge degrees of freedom in the Hubbard
chain become frozen and the system maps onto a ladder where the
nonlocal antiferromagnetic Kondo coupling effectively induces
local ferromagnetic correlations between the spins in the ladder
(vertical dashed lines). (b) The system depicted in (a) can be
mapped onto a S ¼ 1 Haldane chain, which supports topologically protected S ¼ 1=2 end states [38,47,48,52–56].
host symmetry-protected topological spin-1=2 modes at the
boundaries [52–60]. This situation is also very reminiscent
of the case of the ferromagnetic Kondo lattice [61–64].
To see how these spin-1=2 boundary modes emerge in
our low-energy Hamiltonian (46), we consider solutions
~ s Ψa ðxÞ ¼ 0, where Ψa ðxÞ ¼
of the eigenvalue equation H
a a T
ðξR ; ξL Þ is a Majorana spinor, and a ¼ ð0; 1; 2; 3Þ [38].
In matrix form, this equation is
½−ivs τ̂3 ∂ x þ ma τ̂2 Ψa ðxÞ ¼ 0;
ð47Þ
with ma ¼ −3JK m2 m1 =2π, for a ¼ 0 [ma ¼ J K m2 m1 =2π
for a ¼ ð1; 2; 3Þ] and τ̂i the Pauli matrices acting on the
right- or left-moving space. Equation (47) admits solutions
of the form Ψa ðxÞ ∝ exp ð−ma τ̂1 x=vs ÞΨa ð0Þ, and one
would think that, in principle, two normalizable solutions
in the limit x → ∞ could arise: (1) the choice
Ψa ð0Þ ¼ ð1; −1ÞT , with ma < 0, and (2) Ψa ð0Þ ¼ ð1; 1ÞT ,
with ma > 0. However, note that only the last choice is
compatible with the boundary condition (41). Then, the
only physical solution localized around x ¼ 0 corresponds
to the choice ma > 0, which in our case corresponds to
a ¼ ð1; 2; 3Þ:
Ψa ðxÞ ∝ ξa e−ma x=vs
1
;
1
a ¼ ð1; 2; 3Þ;
ð48Þ
with ξa being a localized Majorana fermion. Using the
expression (43) for the smooth part of the spin density, we
can physically associate the presence of the localized
Majorana bound state with a localized spin-1=2 magnetic
edge state. This is consistent with the results in Ref. [26],
where, in the case of a uniform Kondo interaction,
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a magnetic mode with no admixture with charge degrees of
freedom emerges at the boundaries. However, note that the
origin of these edge states is quite different: while in the
mean-field regime they emerge as a consequence of Kondounscreened end spins in the Heisenberg chain, in our
case they are intimately related to the physics of the
Haldane chain.
We now derive a topological invariant to characterize the
presence of the edge states, using a suitable generalization
of the concept of electrical polarization in 1D insulators
[65–70]. We, therefore, focus on the uniform magnetization
Eq. (43). Although the original Hamiltonian has spinrotational symmetry, the Abelian bosonization is not an
explicitly SU(2)-invariant formalism. Therefore, while the
choice of axes is arbitrary due to the spin-rotation symmetry of the problem, once we define the z direction as the
spin-quantization axis, the perpendicular components of
the magnetization M x ðxÞ and M y ðxÞ acquire a more
complicated mathematical form. For that reason, it is
convenient to focus only on the z component of the
symmetric spin density Eq. (43):
X
Mz ðxÞ ¼
Mz1;λ ðxÞ þ M z2;λ ðxÞ;
λ¼L;R
1
¼ − pffiffiffi ∂ x ϕþ ðxÞ:
π
ð49Þ
In the expressionpabove,
we use Eq. (20) and the definition
ffiffiffi
of ϕþ ðxÞ ¼ ð1= 2Þ½ϕ1 ðxÞ þ ϕ2 ðxÞ. We now define the
total magnetic moment along the ẑ axis at one end
(for concreteness,
the left end) of the chain as
pffiffiffi R
mzT ≡ ð1= π Þ 0xb dxMz ðxÞ, where xb is an unspecified
position in the interior of the chain where the magnetization
reaches the value in the bulk. In bosonic language, it
acquires the compact form
1
mzT ¼ − ½ϕþ ðxb Þ − ϕþ ð0Þ:
π
ð50Þ
From the expression of the bosonic Hamiltonian Eq. (25) in
the limit JK → ∞, we see that the system minimizes the
energy in the bulk by pinning the field ϕþ ðxÞ to one of the
degenerate values:
π
ϕþ ðxb Þ ¼ :
2
ð51Þ
On pthe
ffiffiffi other hand, from the definition of ϕ ðxÞ ¼
ð1= 2Þ½ϕ1s ðxÞ ϕ2s ðxÞ and Eq. (13), the boundary condition ϕþ ð0Þ ¼ 0 is obtained. Replacing these values into
Eq. (50), we obtain the following quantized values of the
magnetic moment at the left end:
1
mzT ¼ :
2
ð52Þ
This magnetic moment at the end of the chain is analogous
to the electrical polarization [65–69] or the time-reversal
polarization [70] in 1D insulators. In particular, we note the
close relation between our Eq. (50) and the expressions for
the displacement operator appearing in Eq. (23) of
Ref. [67], and for the time-reversal polarization appearing
in Eq. (4.8) in Ref. [70], both given in bosonic language.
From here, we can define a Z2 topological invariant that
characterizes the topological phase of the KondoHeisenberg chain,
Q ¼ ð−1Þ2mT =π ¼ ei2πmT ;
z
z
ð53Þ
which in the limit of an infinite system L → ∞ is Q ¼ −1
in the topological phase (JK > 0) and Q ¼ 1 in the trivial
phase (JK < 0).
The bosonic representation also provides an alternative
way to demonstrate the existence of magnetic edge modes.
Since none of the degenerate values (51) of ϕþ ðxÞ in the
bulk satisfy the boundary condition at x ¼ 0, we conclude
that a kink excitation necessarily must emerge near the
boundary in order to connect those values: precisely this
kink excitation gives rise to the spin-1=2 end state, upon
use of Eq. (49). We remind the reader that in Sec. III, using
similar arguments, we demonstrate the absence of kink
configurations in the charge field ϕ1c ðxÞ, and the fact that
there are no charge edge states in the ground state.
V. CONCLUSIONS
We study theoretically a model for a topological 1D
Kondo insulator (the 1D Kondo-Heisenberg model coupled
in the p-wave channel, with an on-site Hubbard interaction
U in the conduction band) using the Abelian bosonization
formalism and derive the two-loop RG flow equations for
the system at half filling. Our RG analysis shows that the
system develops a Mott insulating gap at low enough
temperatures, even if U ¼ 0. Moreover, the remaining spin
degrees of freedom are effectively described by a ferromagnetic spin-1=2 ladder, which in turn maps onto a spin-1
Haldane chain with topologically protected spin-1=2 magnetic edge modes. This situation is reminiscent of the
physics of the ferromagnetic Kondo necklace, which also
maps onto the spin-1 Haldane chain [61–64], although in
our case it arises as a result of the nontrivial structure of the
Kondo coupling.
In contrast to three-dimensional bulk topological Kondo
insulators, where the mean-field approximation is well
justified and the system can be effectively described in
terms of noninteracting quasiparticles opening a (renormalized) hybridization gap near the Fermi surface
[17,19,71], in one spatial dimension the presence of strong
quantum fluctuations cannot be ignored, and one is forced
to use different approaches. The Abelian bosonization
method allows us to obtain a description of the 1D TKI,
which is fundamentally different from the mean-field
021017-8
MAGNETIC END-STATES IN A STRONGLY-INTERACTING …
picture. In the first place, the system develops a Mott gap
(instead of a hybridization gap) in the spectrum of charge
excitations when the conduction band is half filled (small
deviations from half filling do not affect this scenario
qualitatively [36]). This Mott gap arises from umklapp
processes at second order in the Kondo interaction.
Physically, this can be understood as a dynamically induced
effective interaction term, which appears at order J 2K in the
conduction band by integrating out perturbatively shorttime spin excitations in the Heisenberg chain. In contrast to
the mean-field description, where the hybridization gap
depends exponentially on the microscopic Kondo
coupling Δc ∼ exp ð−1=J K Þ, the integration of the RG
Eqs. (34)–(36) in the limit J K → 0 results in Δc ∼ J2K .
Our RG analysis indicates that JK is a relevant perturbation
and flows to strong coupling, dominating the physics at low
temperatures. In particular, at temperatures below the Mott
gap, the charge degrees of freedom are frozen and the
system effectively behaves as a ferromagnetic spin-1=2
ladder, which is known to map onto the spin-1 Haldane
chain. Therefore, our work allows us to make an insightful
connection between two a priori unrelated physical models. Interestingly, exploiting this connection, we predict the
existence of topologically protected spin-1=2 edge states.
This seems to correspond to the “magnetic phase” found by
Alexandrov and Coleman [26], which for a uniform Kondo
coupling J K is characterized by Kondo-unscreened spins at
the end of the Heisenberg chain. However, the emergence
of these edge states again corresponds to a very different
mechanism than the one provided by the mean-field theory.
Interestingly, within the bosonization framework, we are
able to obtain a Z2 topological invariant [see Eq. (53)] in
terms of the magnetization at an end of the chain.
Our work opens the possibility to explore the
physics of broken Z2 × Z2 hidden symmetry and the
existence of a nonvanishing
string order parameter
P
Sα
l≤j<m j Sα i ≠ 0
Oαstring ≡ limjl−mj→∞ hSαl e
(which are
m
well-known features of the Haldane phase [52,53,72]) in
the p-wave Kondo-Heisenberg model. In particular, note
the close relation between the Z2 topological invariant (53)
and the string-order parameter in bosonized form [see
Eq. (83) in Ref. [47]].
Furthermore, we reiterate that the model studied here can
be viewed as a nontrivial strongly correlated generalization
of the old Tamm-Shockley model [32,33]. The latter is a
prototypical one-dimensional model that exhibits a topological phase transition and it can be used to construct highdimensional topological band insulators [34]. Likewise, the
strongly correlated topological Kondo-Heisenberg model
could potentially become a building block in constructing
higher-dimensional strongly interacting topological
states—not adiabatically connected to “simple” topological
band insulators. Although the physical realization of the 1D
p-wave Kondo lattice model studied here in solid-state
systems might be quite challenging, our results might have
iπ
PHYS. REV. X 5, 021017 (2015)
direct application to ultracold-atom experiments, where
double-well optical superlattices loaded with atoms in s and
p orbitals have been realized [27,28]. In such systems, one
can imagine the atoms forming ladders where one of the
legs corresponds to the s orbitals and the other to p orbitals
(e.g., see Ref. [35]). The overlap between s and p orbitals
along the rungs vanishes by symmetry, and therefore only
the off-diagonal hopping tsp survives. Next, allowing for an
on-site repulsive Hubbard U 0 interaction in the s-orbital leg
(using, e.g., Feschbach resonances), one can derive an
effective p-wave Kondo lattice model in the limit
tsp =U → 0, introducing a canonical transformation to
eliminate processes at first order in tsp . At half filling,
the s orbitals are effectively described by SU(2) spins and
the Kondo parameter in our Eq. (3) becomes proportional to
JK ∼ t2sp =U. Therefore, the system can be described by the
model described in this work. Finally, we mention in this
context recent experimental results [29–31], which suggest
the existence of a magnetic phase transition and/or suppressed surface charge transport in select samples of
samarium hexaboride (SmB6 —a three-dimensional topological Kondo insulator). It is possible that these phenomena, which remain unexplained at this stage, involve in a
crucial way an interplay between band topology and strong
correlations, which conceivably may lead to the formation
of nontrivial magnetic topological surface modes reminiscent of the edge states found here. In a more general
context, our results might be relevant to other materials that
belong to the same “Haldane universality class,” thanks to
the connection (unveiled in this work) to the ferromagnetic
Kondo lattice model. For example, the organic molecular
compound Mo3 S7 ðdmitÞ3 at two-third filling, a promising
candidate for a quantum spin liquid, has recently been
shown to be a realization of the ferromagnetic Kondo lattice
model at half filling [63,64], and therefore, it should
realize a Haldane phase with magnetic end modes at low
temperatures.
ACKNOWLEDGMENTS
A. M. L. acknowledges support from National Science
Foundation-Joint Quantum Institute-Physics Frontier
Center (NSF-JQ-PFC) and from program Red de
Argentinos Investigadores y Cientícos en el Exterior
(RAICES), Argentina. A. O. D. is partially supported
by Proyecto de Investigación Plurianual (PIP)
11220090100392 of Consejo de Investigaciones
Científicas y Técnicas (CONICET), Argentina. V. G. was
supported by Department of Energy-Basic Energy Sciences
(DOE-BES) DESC0001911 and Simons Foundation. The
authors would like to thank Thierry Giamarchi, Eduardo
Fradkin, Philippe Lecheminant, Edmond Orignac, Dmitry
Efimkin, Xiaopeng Li and Tigran Sedrakyan for useful
discussions and for pointing us relevant references.
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APPENDIX: DYNAMICALLY GENERATED
INTERACTIONS AND DERIVATION OF THE
RENORMALIZATION-GROUP EQUATIONS
Here, we derive an effective action for the system and the
renormalization-group equations [Eqs. (34)–(36)]. The idea
is to show that umklapp processes, which mimic a repulsive
interaction among electrons in the half filled conduction
band, arise at order OðJ2K Þ and open a gap in the charge
sector of the model. To that end, we expand the generating
functional of the system (i.e., the partition function) up to
second order in the coupling constants Gα following
Refs. [73,74],
ZZ
X Gα Z
Z
1 X Gα Gβ
2
2 2 0
0
d rhOα i0 þ
¼ 1−
d rd r hOα ðrÞOβ ðr Þi0 þ ;
2−Δα
Z0
2 α;β a4−Δα −Δβ
jr−r0 j>a
α a
where indices α and β run on 2c, 3, and K. Here, Δα
is the scaling dimension of the operator Oα defined
in Eqs. (31)–(33) (Δ3 ¼ Δ2c ¼ 2, ΔK ¼ 32), Z0 ¼
RQ
−S0 ½ϕi ;θi is the generating function
i¼f1c;g D½ϕi ; θ i e
of the free theory, and the mean values h i0 also
correspond to that theory. This formalism is standard in
the analysis of 1D quantum systems, and has been applied
in several previous works (see, for example, Ref. [75],
where the method is explained in detail).
The third term of the rhs in Eq. (A1) takes the same form
as the second one if we assume that for r → r0 the product
of two operators fulfills the following operator product
expansion property [73,74] :
Oα ðrÞOβ
ðr0 Þ
X γ
¼
Cαβ
γ
0
Oγ ðrþr
2 Þ
0 Δα þΔβ −Δγ
∣r − r ∣
þ more irrelevant operators;
0
0
∶eiλϕL ðzÞ ∶ ∶eiλ ϕL ðz Þ ∶ ¼
where Oγ includes all the operators generated from
each OPE.
Let us focus on the OPE between two OK operators,
which is the most relevant perturbation in the RG sense,
and is precisely the contribution that leads to the umklapp
processes we are trying to describe. To simplify the
discussion, here we return to the representation of the
bosonic fields in terms of left and right movers ϕL ðzÞ and
ϕR ðz̄Þ [see Eq. (8)], with z ¼ vF τ þ ix and z̄ ¼ vF τ − ix.
We now assume to be sufficiently far away from the
boundaries. In these conditions, the boundary conditions
(9) and (10) and the commutation relation (11) can be
effectively neglected, and the fields ϕL ðzÞ and ϕR ðz̄Þ
become independent (i.e., they do not mix). This allows
us to focus on only the processes involving the left-moving
field ϕL ðzÞ (for right-moving fields, we just need to change
L → R and z → z̄). The basic OPE we need is
ðA2Þ
λλ0 =2
2π
1
iλ∂ z0 ϕL ðz0 Þ
0
0
∶eiðλþλ ÞϕL ðz Þ
þ
þ
∶;
0
0
L
ðz − z0 Þ−λλ ðz − z0 Þ−λλ −1
which is obtained by normal ordering the rhs expression
and then developing for z0 near z. Through repeated use of
this expression, we obtain the desired OPE, which reads
3 O3
3 O2c
0 −
4π ∣z − z ∣ 4π ∣z − z0 ∣
3 O−
1 Oþ
þ
;
0 −
4π ∣z − z ∣ 4π ∣z − z0 ∣
ðA1Þ
OK ðz; z̄ÞOK ðz0 ; z̄0 Þ ¼ −
ðA4Þ
1
where we define the operators O ≡ 4π
∂ z̄ ϕR ∂ z ϕL [which
also appear in the z component of the product of the right and
left smooth-varying spin currents, in the first two lines in
Eq. (3.8) of Ref. [45]]. Note that these terms break the SU(2)
invariance of the model. This is a well-known feature of the
Abelian bosonization prescription, which is not explicitly
SU(2)-invariant formalism [36,37]. This means that one has
to keep track of all contributions to recover the SU(2)
invariance and, vice versa, neglecting irrelevant operators
ðA3Þ
[as we do to obtain the action in Eq. (26)] might result in
apparent inconsistencies in the formalism. In our case, this
problem has no consequences for our purposes because the
operators O renormalize the couplings of the marginal
contributions, which we, in any case, neglect in relation to
the relevant contribution ∼∶N1 · N2 ∶. Therefore, we will not
consider these operators.
On the other hand, the first line in Eq. (A4) is physically
more interesting, as the operators O3 and O2c were already
present in the action (28) corresponding to the Hubbard
model. If we insert (A4) into (A1), change variables as
r̂ ¼ r − r0 and R ¼ r þ r0 =2, and integrate over r̂ imposing
a cutoff of order a, we obtain an expression that renormalizes the first-order contribution. We identify the effective
coupling for operators O2c and O3 as
021017-10
3
Ĝ02c ¼ G02c þ G2K ;
8
ðA5Þ
MAGNETIC END-STATES IN A STRONGLY-INTERACTING …
and the same for Ĝ03. Therefore, we show that the interchain
Kondo coupling generates an effective Hubbard repulsion
Ueff ¼ ð3a=8ÞððJK m2 Þ2 =πvF Þ in the conduction chain. The
equation above can be physically interpreted as umklapp
processes (generated by integrating out fast spin fluctuations in the Heisenberg chain at second order in the
interchain Kondo coupling) which mimic the effect of an
interaction in the conduction band.
To determine the actual dependence of the charge gap Δc
with respect to the parameters of the model, we need to
derive the RG flow equations. This is achieved following
similar steps as in the previous paragraphs. The main idea
is that the theory defined with a microscopic cutoff a
should remain invariant under a scaling transformation
a → að1 þ dlÞ, where dl is a dimensionless infinitesimal.
Therefore, the couplings Gα ðaÞ in Eq. (A1) must be
changed in such a way that they preserve the generating
functional, i.e., Z½a ¼ Z½að1 þ dlÞ. The method is standard and we refer the reader to Ref. [73] for details. The
renormalization group flow equations can be written in
terms of the coefficients Cγαβ as
X
dGγ
¼ ð2 − Δγ ÞGγ − π Cγαβ Gα Gβ ;
dl
αβ
ðA6Þ
3
are
where the coefficients C2c
KK ¼ CKK ¼ −ð3=4πÞ
extracted from Eq. (A4). The remaining coefficients are
obtained by the OPEs between the corresponding operators. Following the lines of Ref. [39], we straightforwardly
obtain
1
;
2π
1
¼− ;
8π
C32c 3 ¼ −
CK2c K
1
C2c
33 ¼ − ;
π
1
CK3 K ¼ − :
2π
ðA7Þ
Inserting these values into Eq. (A6), we obtain
Eqs. (34)–(36) in the main text.
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